Determin the absolute extreme values of the function on the given interval.
1.)y= 3 sin x + 4 cos x, xE[0, 2pie]
dy/dx = 3 cos x - 4 sin x
=0 at max or min
3 cos x = 4 sin x
so
tan x = 3/4 at extreme
3,4,5 triangle.
tan is 3/4 in quadrants 1 and 3
In quadrant 1
sin x = 3/5
cos x = 4/5
3 sin x + 4 cos x = 9/5 + 16/5 = 5 (maximum probably)
do similar computation in quadrant 3 and I suspect you will find the minimum.
y'=3cosx-4sinx=0
3cosx=4sinx
tanx=3/4
solve for x. You will get two angles, 180degrees apart.
One is a max, one is a min.
To determine the absolute extreme values of a function on a given interval, we first need to find the critical points of the function. The critical points are the values of x where the derivative of the function is either zero or undefined.
To find the derivative of y = 3sin(x) + 4cos(x), we can apply the chain rule. The derivative of sin(x) with respect to x is cos(x), and the derivative of cos(x) with respect to x is -sin(x). Therefore, the derivative of y is:
dy/dx = 3cos(x) - 4sin(x)
Next, we need to find the critical points by setting the derivative equal to zero:
3cos(x) - 4sin(x) = 0
Now, we can solve for x:
3cos(x) = 4sin(x)
Divide both sides by cos(x):
3 = (4sin(x)) / cos(x)
Using the trigonometric identity tan(x) = sin(x) / cos(x), we can rewrite the equation as:
3 = 4tan(x)
Now, solve for x by taking the inverse tangent of both sides:
x = atan(3/4)
Since the interval is [0, 2pi], we can check the value of y(x) at the boundaries of this interval, which are x = 0 and x = 2pi:
y(0) = 3sin(0) + 4cos(0) = 0 + 4 = 4
y(2pi) = 3sin(2pi) + 4cos(2pi) = 0 - 4 = -4
Now, we evaluate the function at the critical point x = atan(3/4):
y(atan(3/4)) = 3sin(atan(3/4)) + 4cos(atan(3/4))
To find the value of sin(atan(3/4)) and cos(atan(3/4)), we can use the right triangle definition. Let's assume a right triangle with an angle A such that tan(A) = 3/4. If we take the opposite side as 3 and the adjacent side as 4, then the hypotenuse can be found using the Pythagorean theorem:
hypotenuse^2 = 3^2 + 4^2
hypotenuse^2 = 9 + 16
hypotenuse^2 = 25
hypotenuse = 5
Therefore, sin(A) = 3/5 and cos(A) = 4/5.
Plugging these values into the equation:
y(atan(3/4)) = 3(3/5) + 4(4/5) = 9/5 + 16/5 = 25/5 = 5
Now, we compare the values:
y(0) = 4, y(2pi) = -4, y(atan(3/4)) = 5
The absolute maximum value is 5, which occurs at x = atan(3/4), and the absolute minimum value is -4, which occurs at x = 2pi.
Therefore, the absolute extreme values for the function y = 3sin(x) + 4cos(x) on the interval [0, 2pi] are a maximum of 5 at x = atan(3/4) and a minimum of -4 at x = 2pi.